"1uF+1uF at 10 volts in parallel = 2uF at 10volts = 100 uJ
(1/1uF) + (1/1uF) = (1/2uF) or simply 0.5uF at 20 volts (10 +10) in series = 100uJ"

Question. Why falling charge to half if capacitors connected to series?

Because when you put cap in serie the inner plates which hold +Q and -Q get nullified so you have a composite cap with two time the voltage but halved in capacity !!!

And another strange thing:

When we separate the plates of a charged capacitor then its stored electrical energy increases due to the applied mechanical energy. But what happens to the stored energy in the capacitor if we move the plates closer together? Does the energy disappear to nowhere or do we get back the mechanical energy? If so, then what happens if we move the plates closer together of a capacitor which is not charged? Do we get negative energy because moving the plates closer together means less energy from a start point of zero energy?

1) If you move the plate together energy decrease, if the plate touch themselves you just short the cap, energy dissiped...
2) When he is not charged no electrostatic force play with/or against you, you just play with two metal sheets.
3) No negative energy. (see answer one).


When cap is charged the electrostatic force pull the plate together, that's why energy decrease.
When you push plates apart you invest energy in that cap, you convert Force*Distance to electrical energy !
SRM.

When you push plates apart you invest energy in that cap, you convert Force*Distance to electrical energy !

That's very strange stuff. Because it means that the plates of a capacitor act like a tension spring. The more I pull on a tension spring the more energy is stored in that spring (till I release it). The more I push the plates of a capacitor apart the more energy is stored in that capacitor. But what if I push the plates a few miles apart? Could this increase the energy in the capacitor to infinity (if I push one plate to the other end of the universe)? And what will happen if I release the plates?

This sounds quite not complicated enough. Therefore I'm daring to ask what could happen (in terms of differential equations) if both coils would share the same core? What could happen if the coils are bifilar wound? What could happen if the core goes into saturation? What could happen if these are air coils without a ferromagnetic core?

I need more info to answer that, e.g.:
Are the two mutually-coupled coils "charged" sequentially or simultaneously? 
Is the bifilar winding wound in bucking mode or aiding mode (BTW: this detail should always be specified when using the phrase "bifilar winding") ? 
Does the saturation refer to a soft ferromagnetic core or a hard ferromagnetic core (hard ones exhibit high magnetic remanance after the magnetizing H field is removed) ?

But anyway I would not rely too much on differential equations because something must be wrong with these equations, otherwise there would be neither the presentation of the Kapanadze device nor of the Stepanov transformer nor of the Steven Marks TPU (not mentioned the story of Tesla's electric car). So we should keep looking for the one differential equation that is wrong or missing.

These equations are based on centuries of empirical observations, such as voltage measured across inductors when the magnetic field inside them changes or the current that flows in a capacitance/resistance in response to voltage.
To invalidate those equations one of the effects observed by Ampere, Faraday, Lenz, Lorentz, Biot-Savart, Maxwell, Ohm would have to be proved to be wrong or incomplete or nonexistent.  The former two are still possible but require extraordinary experimental proof.

The second argument that Kapanadze's or Stepanov devices work contrary to these equations is a weak one.  First of all we cannot be sure that they work at all.  Secondly we do not know their operating principle - for example Ampere, Faraday, Lenz, Lorentz, Biot-Savart, Maxwell, Ohm laws might be simply not applicable to their operational principle and the equations, that embody those effects, likewise might not be applicable.  For instance: McFreey's analysis does not violate any of the aforementioned effects and equations that stem from them, the electronic schematic of his system looks like an ordinary transformer yet it allegedly generates copious amounts of energy through a completely different effect - different from the operation of a conventional transformer.

By the same token, it is entirely possible that entirely different types of electricity exist (e.g. uncharged electrons, cold electricity,  birotating electrons - Cooper pairs, electron triplets in counterspace) and that these different types do not obey the laws of conventional electricity.
However, merely stating the possibility does not make it so.  It should be possible to discriminate between the conventional electricity and the new types of electricity, somehow.  This difference should be demonstrable, even if it is one measly effect that the conventional electricity is not capable of exhibiting (e.g. cooling resistors, etc...). 
This should be a simple effect, that everyone can replicate, not high-speed data transmission from Miami to Havana without an antenna. 
Anyway, as usual, extraordinary claims require extraordinary proof.

 The resistance of that wire you use to collect the energy difference would eat up your gains.


 Lets think about this in a different way. What do you think a body in space is? A capacitance? Each planet in a solar system is one terminal of a capacitor. The main capacitance is the central point like our sun. Now measure the differences between each planet and the sun. The sun us the positive plate with each plate becoming a portion of the negative plate. Can you say gravity.... Gravity is a misnomer in space and reaches out for light years. Each capacitance set, sun and planet set, interacts with each other but are always in balance. This is the effect we call gravity.

so I'm what a postivily charged particle that sticks to the earth?  electrostatic attractive force != gravity.

At least you are thinking for yourself.
Try to enumerate all of the known differences between electrostatic and gravitational attraction, for the benefit of other users on this forum.

I figured one would be enough, but to extend that one a little, then every other person is also positive, which would make shaking hands or hugging difficult.

 No no no. The attractive force is on charges that are freely flowing around our globe. These charges when traveling by us and onward to our core or central point cause a gravity like effect upon everything around the globe and even outside of our global atmosphere. When that charges get to the core int they are converted into a magnetic flux by the material surrounding the core point. The charges, now transformed into a magnetic flux exit the core out through our poles and then recycle back into the earth via the atmosphere. There are steps I'm not showing as well like heat buildup in the core from the many charges flowing in the core area. Some of these are radiated from the core in other well know processes like convection. It's a very complicated process we are defining here.
 So gravity is actually the force upon our own charges as the flow of free charges goes by us towards the core. Although we do have a charge value and charges in our matter they are encapsulated in little capacitors we like to call cells.
 In the earth, if we understand the static laws and apply them we understand that the charges are drawn away from the surface of the earth and back down in the mid section of the globe. Now some do get away and that's the natural heat we see radiating away from the earth.


 Lets look at the Tesla experiment of an elevated conductor above our planet and you will see that the elevated conductor is one plate of a capacitor and the other is the ground surface. The further these plates are the higher the voltage potential across that crude capacitor. We all know there is an electric field around our planet and this fact creates a flow direction. Since the earths crust is charge deficient it has a vacuum associated with it. This causes a flow direction from the elevated conductor towards the earths plate when it is referenced to the grounds plate and it will cause a good amount of charges to bypass the normal avenue of flow directly to the earths crust plate.

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Ya, unfortunately I don't really subscribe to the electric universe theory; electrostatic charge is so much weaker than magnetism.  But then I think the earth's core moves because of the magnetic field it's in, not that the field exists because of a dynamo motion.


  Primer Fields.  Though; this doesn't account for why the fields are there in the first place on a large scale.  He's got an interesting model, but it's lacking also.  When voyager 1 and 2 left the heliosphere they encountered even larger magnetic fields.  Acutually this clip accounts for both opposing and attractive forces on the [size=78%]planets/sun.  [/size]